Question

Eigenvectors of a=[1 1 1; 1 1 1; 1 1 1]

Answer 1

The eigenvectors of the matrix [1 1 1; 1 1 1; 1 1 1] can be found by solving the equation (a - λI)v = 0, where λ is the eigenvalue and I is the identity matrix.

Step-by-step explanation:

In linear algebra, the eigenvectors of a matrix are the vectors that do not change their direction when multiplied by the matrix. To find the eigenvectors of the matrix a = [1 1 1; 1 1 1; 1 1 1], we need to solve the equation (a - λI)v = 0, where λ is the eigenvalue and I is the identity matrix. Let's solve it step by step:

1. First, subtract λ from the diagonal elements of matrix a, yielding [1-λ 1 1; 1 1-λ 1; 1 1 1-λ].
2. Next, write out the resulting system of equations: (1-λ)x + y + z = 0, x + (1-λ)y + z = 0, x + y + (1-λ)z = 0.
3. Simplify each equation: x + y + z = λx, x + y + z = λy, x + y + z = λz.

From the simplified equations, we can see that any non-zero vector that satisfies x + y + z = 0 will be an eigenvector of matrix a. These eigenvectors form a one-dimensional subspace called the eigenspace associated with the eigenvalue 0.